Pythagoreanism originated in the 6th century BCE, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics and mysticism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised a marked influence on Aristotle, and Plato, and through them, all of Western philosophy.
Aristotle makes clear that there are several groups of people included under the heading “so-called Pythagoreans,” by explicitly distinguishing those Pythagoreans who posited the table of opposites from the main Pythagorean system.
According to tradition, pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi (μαθηματικοί, Greek for "Teachers") and the akousmatikoi (ἀκουσματικοί, Greek for "listeners"). There is the inner and outer circle
John Burnet (1892) noted
Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. And it will be found that the hypothesis, if it is to be called by that name, of a regular organisation of scientific activity will alone explain all the facts. The development of doctrine in the hands of Thales, Anaximander, and Anaximenes, for instance, can only be understood as the elaboration of a single idea in a school with a continuous tradition.
There were also two forms of philosophy, for the two genera of those that pursued it: the Acusmatici and the Mathematici. The latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras but from Hippasus. The philosophy of the Acusmatici consisted in auditions unaccompanied with demonstrations and a reasoning process; because it merely ordered a thing to be done in a certain way and that they should endeavor to preserve such other things as were said by him, as divine dogmas. Memory was the most valued faculty. All these auditions were of three kinds; some signifying what a thing is; others what it especially is, others what ought or ought not to be done. (p. 61)
The best of all legislators came from the school of Pythagoras, Charondas, the Catanean, Zaleucus and Timaratus as well as many others, who established laws with great benevolence and political science. (p. 26)
The whole Pythagoric school produced appropriate songs, which they called exartysis or adaptations; synarmoge or elegance of manners and apaphe or contact, usefully conducting the dispositions of the soul to passions contrary to those which it before possessed. By musical sounds alone unaccompanied with words they healed the passions of the soul and certain diseases, enchanting in reality, as they say. It is probable that from hence this name epode, i. e., "enchantment," came to be generally used. For his disciples, Pythagoras used divinely contrived mixtures of diatonic, chromatic and enharmonic melodies, through which he easily transferred and circularly led the passions of the soul in a contrary direction, when they had recently and in an irrational and secret manner been formed; such as sorrow, rage and pity, absurd emulation and fear, all-various desires, angers and appetites, pride, supineness and vehemence. Each of these he corrected through the rule of virtue, attempering them through appropriate melodies, as well as through certain salubrious medicine. (p.43)
Excerpt from a speech by the character ‘Aristotle’ in Protrepticus (Hutchinson and Johnson, 2015)
Thus nothing is more valuable than intelligence, which we say is a capacity of the most authoritative thing in us, to judge one disposition in comparison with another, for the cognitive part, both apart and in combination, is better than all the rest of the soul, and knowledge is its virtue. (p. 35)
Therefore its function is none of what are called ‘parts of virtue’, for it is better than all of them and the end produced is always better than the knowledge that produces it. Nor is every virtue of the soul in that way a function, nor is success; for if it is to be productive, different ones will produce different things, as the skill of building (which is not part of any house) produces a house. However, intelligence is a part of virtue and of success, for we say that success either comes from it or is it. (p. 36)
According to historians like Thomas Gale (based on Archytas account), Thomas Taler (based on the work of Iamblichus), or Cantor, Archytas (428 BC) became the head of the school, about a century after the murder of Pythagoras. According to August Böckh (1819), who cites Nicomachus, Philolaus was the successor of Pythagoras. And according to Cicero (de Orat. III 34.139), Philolaus was teacher of Archytas of Tarentum.
According to the historian's from the Stanford Encyclopedia of Philosophy, "Philolaus and Eurytus are identified by Aristoxenus as teachers of the last generation of Pythagoreans" (D. L. VIII 46).
The mathēmatikoi were supposed to have extended and developed the more mathematical and scientific work begun by Pythagoras. The mathēmatikoi did think that the akousmatikoi were Pythagorean, but felt that their own group was more representative of Pythagoras.
Excerpt from a speech by the character ‘Aristotle’ in Protrepticus (Hutchinson and Johnson, 2015 p. 32)
Many of the more recent Pythagoreans assumed that mathematics has as its subject matter only the things that are the same and in the same way, and hypothesized only these principles; so in the same way they define as different both the sciences and the demonstrations about such things. But since both in the speeches preceding this point and in the later remarks we will demonstrate that there are many and different substances that are unchangeable and exist in the same state, not only the ones in mathematics, and that those are more senior and more honorable than these, and we will also demonstrate that these mathematical principles are not the only ones, but there are also others, and these in fact are more senior and more powerful than those, and that these are not the principles of all the things that exist but only of some; so it is for these reasons that the mathematical demonstration now demands a determination of which of the qualities it can demonstrate remain the same and in the same way, and from what kinds of principles it reasons, and about what kinds of problems it produces the demonstrations.
Commentary from Sir William Smith, Dictionary of Greek and Roman Biography and Mythology (1870, p. 620).
Pythagoras resembled greatly the philosophers of what is termed the Ionic school, who undertook to solve by means of a single primordial principle the vague problem of the origin and constitution of the universe as a whole. But, like Anaximander, he abandoned the physical hypotheses of Thales and Anaximenes, and passed from the province of physics to that of metaphysics, and his predilection for mathematical studies led him to trace the origin of all things to number, this theory being suggested, or at all events confirmed, by the observation of various numerical relations, or analogies to them, in the phenomena of the universe. "Since of all things numbers are by nature the first, in numbers they (the Pythagoreans) thought they perceived many analogies to things that exist and are produced, more than in fire, and earth, and Avater; as that a certain affection of numbers was justice; a certain other affection, soul and intellect; another, opportunity; and of the rest, so to say, each in like manner; and moreover, seeing the affections and ratios of what pertains to harmony to consist in numbers, since other things seemed in their entire nature to be formed in the likeness of numbers, and in all nature numbers are the first, they supposed the elements of numbers to be the elements of all things". Brandis, who traces in the notices that remain more than one system, developed by different Pythagoreans, according as they recognised in numbers the inherent basis of things, or only the patterns of them, considers that all started from the common conviction that it was in numbers and their relations that they were to find the absolutely certain principles of knowledge, and of the objects of it, and accordingly regarded the principles of numbers as the absolute principles of things; keeping true to the common maxim of the ancient philosophy, that like takes cognisance of like. Aristotle states the fundamental maxim of the Pythagoreans in various forms.
According to Philolaus, number is the "dominant and self-produced bond of the eternal continuance of things." But number has two forms, the even and the odd, and a third, resulting from the mixture of the two, the even-odd. This third species is one itself, for it is both even and odd. One, or unity, is the essence of number, or absolute number, and so comprises these two opposite species. As absolute number it is the origin of all numbers, and so of all things. (According to another passage of Aristotle, Met. xii. 6. p. 1080, b. 7. number is produced) This original unity they also termed God (Ritter, Gesch. der FML vol. i. p. 389). These propositions, however, would, taken alone, give but a very partial idea of the Pythagorean system. A most important part is played in it by the ideas of limit, and the unlimited. They are, in fact, the fundamental ideas of the whole. One of the first declarations in the work of Philolaus was, that all things in the universe result from a combination of the unlimited and the limiting; for if all things had been unlimited, nothing could have been the object of cognizance. From the unlimited were deduced immediately time, space, and motion (Stob. Eel. Phys. p. 380 ; Simplic. iii Arist. Phys. f. 98, b.; Brandis, I.e. p. 451). Then again, in some extraordinary manner they connected the ideas of odd and even with the contrasted notions of the limited and the unlimited, the odd being limited, the even unlimited.
According to Sir William Smith, (1870).
The Orphic worshippers of Bacchus did not indulge in unrestrained pleasure and frantic enthusiasm, but rather aimed at an ascetic purity of life and manners. (See Lobeck, Aglaoph. p. 244.) The followers of Orpheus, when they had tasted the mystic sacrificial feast of raw flesh torn from the ox of Dionysus, partook of no other animal food. They wore white linen garments, like Oriental and Egyptian priests, from whom, as Herodotus remarks, much may have been borrowed in the ritual of the Orphic worship. Herodotus not only speaks of these rites as being Egyptian, but also Pythagorean in their character. The explanation of this is that the Pythagorean societies, after their expulsion from Magna Graecia, united themselves with the Orphic societies of the mother country, and of course greatly influenced their character.
But before this time the Orphic system had been reduced to a definite form by Pherecydes and Onomacritus, who stand at the head of a series of writers, in whose works the Orphic theology was embodied; such as Cercops, Brontinus, Orpheus of Camarina, Orpheus of Croton, Arignote, Persinus of Miletus, Timocles of Syracuse, and Zopyrus of Heracleia or Tarentum (Mliller, p. 235). Besides these associations there were also an obscure set of mystagogues derived from them, called Orpheotelests ('OpipeorcAeaTal), "who used to come before the doors of the rich, and promise to release them from their own sins and those of their forefathers, by sacrifices and expiatory songs; and they produced at this ceremony a heap of books of Orpheus and Musaeus, upon which they founded their promises" (Platon Ion, p. 536, b.; Muller, p. 235). The nature of the Orphic theology, and the points of difference between it and that of Homer and Hesiod, are fully discussed by Muller (Hist. Lit. Anc. Gr. pp. 235—238) and Mr. Grote (vol. i. pp. 22, & c.); but most fully by Lobeck, in his Aylaophamus.
Aristotle says the poet Orpheus never existed; the Pythagoreans ascribe this Orphic poem to a certain Cercon.
It appears, in fact, from this, as well as from the extant fragments, that the first book (from Philolaus) of the work contained a general account of the origin and arrangement of the universe. The second book appears to have been an exposition of the nature of numbers, which in the Pythagorean theory are the essence and source of all things. (p. 305)
Excerpt from a speech by the character ‘Aristotle’ in the book Protrepticus (Hutchinson and Johnson, 2015 p. 16)
..the Pythagoreans honored the effort put into mathematics, and coordinated it with the observation of the cosmos in various ways, for example: by including number in their reasoning from the revolutions and their difference between them, by theorizing what is possible and impossible in the organization of the cosmos from what is mathematically possible and impossible, by conceiving the heavenly cycles according to commensurate numbers with a cause, and by determining measures of the heaven according to certain mathematical ratios, as well as putting together the natural science which is predictive on the basis of mathematics, and putting the mathematical objects before the other observable objects in the cosmos, as their principles.
Pythagorean thought was dominated by mathematics. In the area of cosmology there is less agreement about what Pythagoras himself actually taught. The Pythagorean conception of substance, on the other hand, is of unknown origin, partly because various accounts of his teachings are conflicting. The Pythagorean account actually begins with Anaximander's teaching that the ultimate substance of things is "the boundless," or what Anaximander called the "apeiron." The Pythagorean account holds that it is only through the notion of the "limit" that the "boundless" takes form. (See also Philolaus)
Pythagoras wrote nothing down, and relying on the writings of Parmenides, Empedocles, Philolaus and Plato (people either considered Pythagoreans, or whose works are thought deeply indebted to Pythagoreanism) results in a very diverse picture in which it is difficult to ascertain what the common unifying Pythagorean themes were. Relying on Philolaus, whom most scholars agree is highly representative of the Pythagorean school, one has a very intricate picture. Aristotle explains how the Pythagoreans (by which he meant the circle around Philolaus) developed Anaximander's ideas about the apeiron and the peiron, the unlimited and limited, by writing that:
... for they [the Pythagoreans] plainly say that when the one had been constructed, whether out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited by the limit.
Continuing with the Pythagoreans:
The Pythagoreans, too, held that void exists, and that it enters the heaven from the unlimited breath – it, so to speak, breathes in void. The void distinguishes the natures of things, since it is the thing that separates and distinguishes the successive terms in a series. This happens in the first case of numbers; for the void distinguishes their nature.
When the apeiron is inhaled by the peiron it causes separation, which also apparently means that it "separates and distinguishes the successive terms in a series." Instead of an undifferentiated whole we have a living whole of inter-connected parts separated by "void" between them. This inhalation of the apeiron is also what makes the world mathematical, not just possible to describe using maths, but truly mathematical since it shows numbers and reality to be upheld by the same principle. Both the continuum of numbers (that is yet a series of successive terms, separated by void) and the field of reality, the cosmos — both are a play of emptiness and form, apeiron and peiron. What really sets this apart from Anaximander's original ideas is that this play of apeiron and peiron must take place according to harmonia (harmony).
Work and theories
After attacks on the Pythagorean meeting-places at Croton, the movement dispersed, but regrouped in Tarentum, also in Southern Italy. A collection of Pythagorean writings on ethics collected by Taylor show a creative response to the troubles.
The legacy of Pythagoras, Socrates and Plato was claimed by the wisdom tradition of the Hellenized Jews of Alexandria, on the ground that their teachings derived from those of Moses. Through Philo of Alexandria this tradition passed into the Medieval culture, with the idea that groups of things of the same number are related or in sympathy. This idea evidently influenced Hegel in his concept of internal relations.
According to Kirk and Raven (1956), the ancient Pythagorean pentagram was drawn with two points up and represented the doctrine of Pentemychos. Pentemychos means "five recesses" or "five chambers," also known as the pentagonas — the five-angle, and was the title of a work written by Pythagoras' teacher and friend Pherecydes of Syros.
Pythagoreans distinguished three kinds of lives: Theoretic, Practical and Apolaustic. Pythagoras is said to have used the example of Olympic games to distinguish between these three kind of lives.
- It remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape. As to its position there is some difference of opinion. Most people–all, in fact, who regard the whole heaven as finite–say it lies at the centre. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. They further construct another earth in opposition to ours to which they give the name counterearth.
According to Stobaeus his theory was not that the earth revolved around the sun (Heliocentrism), but that it revolved around a hypothetical astronomical object he called the Central Fire, around which the sun also revolved. This system has been called "the first coherent system in which celestial bodies move in circles".
Revolving around the Central Fire above Earth were the Moon, the Sun, the planets, and finally—perhaps fixed and not rotating at all—were the stars. Revolving around the Central Fire below Earth was another hypothetical astronomical object, the Counter-Earth. Whether Philolaus believed Earth to be round or flat—there is "no explicit statement about the shape of the earth in Philolaus' system"—he did not believe the earth rotated, so that the Counter-Earth and the Central Fire were both not visible from Earth's surface—or at least not from the hemisphere where Greece was located.
Music and harmony
Aristoxenus said that music was used to purify the soul just like medicine was used to purge the body. He also wrote several books about Pythagorean, a book on choruses (Περὶ χορῶν): fr. 103 Wehrli, on tragic dancing (Περὶ τραγικῆς ὀρχήσεως): fr. 104-106 Wehrli, and comparisons of dances (Συγκρίσεις): fr. 109 Wehrli, or on Listening to Music, on Tonoi, on Auloi and Instruments, as well as Elementa harmonica.
Related teachings were recorded by Philolaus' pupil Archytas in a lost work entitled On Harmonics or On Mathematics, and this is the influence that can be traced in Plato.
Commentary from Sir William Smith, Dictionary of Greek and Roman Biography and Mythology (1870, p. 620).
- Musical principles played almost as important a part in the Pythagorean system as mathematical or numerical ideas. The opposite principia of the unlimited and the limiting are, as Philolaus expresses it neither alike, nor of the same race, and so it would have been impossible for them to unite, had not harmony stepped in."This harmony, again, was, in the conception of Philolaus, neither more nor less than the octave (Brandis, /. c. p. 456). On the investigation of the various harmonical relations of the octave, and their connection with weight, as the measure of tension, Philolaus bestowed considerable attention, and some important fragments of his on this subject have been preserved. We find running through the entire Pythagorean system the idea that order, or harmony of relation, is the regulating principle of the whole universe.
- ...the hammers beating out a piece of iron on the anvil and producing sounds that accorded with each other, one combination only excepted. In these sounds he recognized the diapason, the diapente and the diatessaron, harmony. And the sound that was between the diatessaron and the diapente was by itself dissonant, yet gave completion to that which was the greater sound among them. (p.49)
A musical scale presupposes an unlimited continuum of pitches, which must be limited in some way in order for a scale to arise. The crucial point is that not just any set of limiters will do. One may not simply choose pitches at random along the continuum and produce a scale that will be musically pleasing. The diatonic scale, also known as "Pythagorean," is such that the ratio of the highest to the lowest pitch is 2:1, which produces the interval of an octave. That octave is in turn divided into a fifth and a fourth, which have the ratios of 3:2 and 4:3 respectively and which, when multiplied, make an octave. If we go up a fifth from the lowest note in the octave and then up a fourth from there, we will reach the upper note of the octave. Finally the fifth can be made up by multiplication of three (largest) whole tones (each corresponding to the ratio of 9:8) and a perfecting semitone (with a ratio of 256:243). Likewise, the fourth can be made up of two whole tones and the same perfecting semitone. This is a good example of a concrete applied use of Philolaus’ reasoning. In Philolaus' terms the fitting together of limiters and unlimiteds involves their combination in accordance with ratios of numbers (harmony). Similarly the cosmos and the individual things in the cosmos do not arise by a chance combination of limiters and unlimiteds; the limiters and unlimiteds must be fitted together in a "pleasing" (harmonic) way in accordance with number for an order to arise.
- The chorus of the Muses was always one and the same, and they had charge of unison, harmony and rhythm, all that goes to make up concord.
A journal review in 1938 mentioned that Boyancé's book was about Muses among the Greek philosophers, and mentions Orpheus as a musician. A thesis of the book is that Pythagoras had at the outset two things: dance and music, and secondly, the use of numbers.
Excerpt from a speech by the character ‘Aristotle’ in Protrepticus (Hutchinson and Johnson, 2015)
- Thus this is what it is to use anything: if the capacity is for a single thing, when someone is doing this very thing, and if the capacity is for a number of things, when he is doing the best of them; for example, with flutes, one uses them either only when playing the flute, or most of all then, as its other uses are presumably also for the sake of this. Thus one should say that someone who uses a thing correctly is using it more, for the natural objective and mode of use belong to someone who uses a thing in a beautiful and precise way. And now the only function of the soul, or else the greatest one of all, is thinking and reasoning. Therefore it is now simple and easy for anyone to draw the conclusion that he who thinks correctly is more alive, and he who most attains truth lives most, and this is the one who is intelligent and observing according to the most precise knowledge; and it is then and to those that living perfectly, surely, should be attributed, to those who are using their intelligence, i.e. to the intelligent. (p. 56)
Some stories of Pythagoras' murder revolve around his aversion to beans. According to legend, enemies of the Pythagoreans set fire to Pythagoras' house, sending the elderly man running toward a bean field, where he halted, declaring that he would rather die than enter the field – whereupon his pursuers slit his throat. It has been suggested that the prohibition of beans was to avoid favism; susceptible people may develop hemolytic anemia as a result of eating beans, or even of walking through a field where bean plants are in flower. It is more likely to have been for magico-religious reasons, perhaps because beans obviously demonstrate the potential for life, perhaps because they resemble the kidneys and genitalia. There was a belief that beans and human beings were created from the same material.
According to accounts from Diogenes Laertius, and or Eustathius, it is thought that the fava bean was particularly sacred to the Pythagoreans; this is because fava beans have hollow stems, and it was believed that souls of the deceased would travel through the ground, up the hollow stems, into the beans where they would reside.
Women were given equal opportunity to study as Pythagoreans, and learned practical domestic skills in addition to philosophy. Women were held to be different from men, sometimes in positive ways. The priestess, philosopher and mathematician Themistoclea is regarded as Pythagoras' teacher; Theano, Damo and Melissa as female disciples. Pythagoras is also said to have preached that men and women ought not to copulate during the summer, holding that winter was the appropriate time.
Female Pythagoreans are also some of the first female philosophers from which we have texts. Although it is debated as to whether or not all of the texts we have were actually written by women, we can see where Pythagoreanism left room for women to practice philosophy within their tradition.
Female philosophers include: Pythais (mother of Pythagoras), Theano (wife of Pythagoras), Cheilonis, Tyrsenis, Myia (Daughter of Theano and Pythagoras), Damo, Timycha, Bitale, Aspasia, Alexis, Perictione (believed to be Plato's mother), Arete, Melissa, Phintys, Ptolemais, Arignote.
Neopythagoreanism was a revival in the 2nd century BC – 2nd century AD period of various ideas traditionally associated with the followers of Pythagoras, the Pythagoreans. Notable Neopythagoreans include Nigidius Figulus, Apollonius of Tyana and Moderatus of Gades. Middle and Neo-Platonists such as Numenius and Plotinus also showed some Neopythagorean influence.
They emphasized the distinction between the soul and the body. God must be worshipped spiritually by prayer and the will to be good. The soul must be freed from its material surroundings by an ascetic habit of life. Bodily pleasures and all sensuous impulses must be abandoned as detrimental to the spiritual purity of the soul. God is the principle of good; Matter the groundwork of Evil. The non-material universe was regarded as the sphere of mind or spirit.
In 1915, a subterranean basilica where 1st century Neo-Pythagoreans held their meetings was discovered near Porta Maggiore on Via Praenestina, Rome. The groundplan shows a basilica with three naves and an apse similar to early Christian basilicas that did not appear until much later, in the 4th century. The vaults are decorated with white stuccoes symbolizing Neopythagorean beliefs but its exact meaning remains a subject of debate.
John Burnet (1892) noted
- The Neoplatonists were quite justified in regarding themselves as the spiritual heirs of Pythagoras; and, in their hands, philosophy ceased to exist as such, and became theology. And this tendency was at work all along; hardly a single Greek philosopher was wholly uninfluenced by it. Perhaps Aristotle might seem to be an exception; but it is probable that, if we still possessed a few such "exoteric" works as the Protreptikos in their entirety, we should find that the enthusiastic words in which he speaks of the "blessed life" in the Metaphysics and in the Ethics (Nicomachean Ethics) were less isolated outbursts of feeling than they appear now. In later days, Apollonios of Tyana showed in practice what this sort of thing must ultimately lead to. The theurgy and thaumaturgy of the late Greek schools were only the fruit of the seed sown by the generation which immediately preceded the Persian War.
- The Pythagorean idea that whole numbers and harmonic (euphonic) sounds are intimately connected in music, must have been well known to lute-player and maker Vincenzo Galilei, father of Galileo Galilei. While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments can be mathematically quantified and described. This may have paved the way to his son's crucial insight that all physical phenomena may be described quantitatively in mathematical language (as physical "laws"), thus beginning and defining the era of modern physics.
- Pythagoreanism has been regarded by some to have had a clear influence on the texts found in the hermetica and thus to have flown over into hermeticism, gnosticism and alchemy.
- The Pythagorean cosmology also inspired the Arab gnostic Monoimus to combine this system with monism and other things to form his own cosmology.
- The pentagram (five-pointed star) has been thought to be an important religious symbol used by the Pythagoreans.
- The Pythagorean school doubtless had a monumental impact on the development of numerology and number mysticism, an influence that still resonates today. For example, it is from the Pythagoreans that the number 3 acquires its modern reputation as the noblest of all digits.
- The Pythagoreans were advised to "speak the truth in all situations," which Pythagoras said he learned from the Magi of Babylon.
- The Pythagorean theory of harmonic ratios is the basis of studies on music theory in the Islamic world, for example al-Farabi's Kitab al-Musiqa al-kabir.
- Pythagorean philosophy had a marked impact on the thoughts of early modern scholars involved within the Scientific Revolution. Of particular interest is the focus applied to the Platonic Solids derived from the Pythagorean theories of geometry and numbers by Plato. Within the work of Leonardo fascination can be found within manuscripts describing the Platonic Solids, and also within the work of Kepler who supported the Copernican theory of heliocentrism and attempted a theory of the universe based on musical, geometrical harmony.
- Dyad (Greek philosophy)
- Esoteric cosmology
- Incommensurable magnitudes
- Mathematical Beauty
- Orphism (religion)
- Pythagorean tuning
- Sacred geometry
- Unit-point atomism
- Stanford Encyclopedia of Philosophy. "Philolaus". Retrieved 30 May 2015.
- John Burnet (1892). Early Greek Philosophy. p. 29.
- Iamblichus (1918). The life of Pythagoras.
- D. S. Hutchinson and Monte Ransome Johnson (25 January 2015). "New Reconstruction, includes Greek text".
- Walter William Rouse Ball (2013). A Short Account of the History of Mathematics.
- August Böckh (1819). Philolaos des Pythagoreers Lehren nebst den Bruchstücken seines Werkes. p. 14.
- Sandra Peterson (2011). "Socrates and Philosophy in the Dialogues of Plato".
- On the two schools and these differences, see Charles Kahn, p. 15, Pythagoras and the Pythagoreans, Hackett 2001.
- Sir Smith William (1870). Dictionary of Greek and Roman biography and mythology. p. 620.
- Sir Smith William (1870). Dictionary of Greek and Roman biography and mythology. pp. 41–42.
- Aristotle; Ross, W. D. (William David), 1877; Smith, J. A. (John Alexander), 1863-1939 (1908). The works of Aristotle. p. 80.
- Sir William Smith (1870). Dictionary of Greek and Roman biography and mythology. p. 305.
- This is actually a lost book whose contents are preserved in Damascius, De principiis, quoted in Kirk and Raven, The Pre-Socratic Philosophers, Cambridge Univ. Press, 1956, page 55.
- Philolaus, Stanford Encyclopedia of Philosophy.
- Aristotle. "13". On the Heavens. II. Retrieved Apr 17, 2016.
- "The Pythagoreans". University of California Riverside. Retrieved 2013-10-20.
- Burch 1954: 272–273, quoted in Philolaus, Stanford Encyclopedia of Philosophy.
- One or more of the preceding sentences incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Stobaeus, Joannes". Encyclopædia Britannica. 25 (11th ed.). Cambridge University Press. p. 929.
- Iamblichus (1918). The life of Pythagoras. p. 49.
- Gillian Clark (1989). Iamblichus: On the Pythagorean Life. p. 18.
- "Reviewed Work: Le culte des Muses chez les philosophes grecs: Etudes d'histoire et de psychologie religieuses by Pierre Boyance". The American Journal of Philology. 1938. doi:10.2307/291187. JSTOR 291187.
- See for instance the popular treatise by Antonio Cocchi, Del vitto pitagorico per uso della medicina, Firenze 1743, which initiated a debate on the "Pythagorean diet".
- Seife, p 38
- Favism, Britannica.com.
- Gabrielle Hatfield, review of Frederick J. Simoons, Plants of Life, Plants of Death, University of Wisconsin Press, 1999. ISBN 0-299-15904-3. In Folklore 111:317-318 (2000).
- Seife, Charles. Zero p 26
- Riedweg, Christoph, Pythagoras: his life, teaching, and influence. Ithaca: Cornell University Press, pp. 39, 70. (2005), ISBN 0-8014-4240-0
- According to Aristotle, the bean plant's unsegmented hollow stem was conceived as "gates of Hades" through which souls migrate back from the underworld to new life in the sunlight. (see Eustathius, In Hom. Il. XIII, 589 See also Diogenes Laertius, "Life Of Pythagoras" XIX).
- Philosophical Misadventures, archived at the Wayback Machine, 24 June 2009, citing Cicero, On Divination, I xxx 62 and Barnes, Jonathan, Early Greek Philosophy, 2nd ed., London: Oxford, 2001. pp. 165–66.
- Glenn, Cheryl, Rhetoric Retold: Regendering the Tradition from Antiquity Through the Renaissance. Southern Illinois University, 1997. 30–31.
- Seife, Charles. Zero p. 27
- Pomeroy, Sarah B. (2013). Pythagorean Women. Baltimore, MD: The Johns Hopkins University Press. p. xiii-xiv. ISBN 9781421409573.
- One or more of the preceding sentences incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Neopythagoreanism". Encyclopædia Britannica. 19 (11th ed.). Cambridge University Press. p. 378.
- Ball Platner, Samuel. "Basilicae". penelope.uchicago.edu.
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- Zammattio, Carlo, "Leonardo The Scientist." Maidenhead, England: Mcgraw-Hill Book Company, 1980. p.98-99
- Koestler, Arthur, "The Sleepwalkers." London, England: Penguin Books, 1959. p.250-251
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